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QuestionConvert the repeating decimals to simplest form:
1. −0.18‾=-0.\overline{18}=
2. 4.45‾=4.\overline{45}=

Studdy Solution

STEP 1

Assumptions1. The numbers given are repeating decimals. . We want to convert these repeating decimals into fractions, which is their simplest form.

STEP 2

Let's start with the first number, −0.18‾-0.\overline{18}. We will denote this number as xx.
x=−0.18‾x = -0.\overline{18}

STEP 3

To get rid of the decimal, we multiply xx by100, because there are two digits in the repeating part.
100x=−18.18‾100x = -18.\overline{18}

STEP 4

Now, we subtract the original equation from this new equation to get rid of the repeating decimal.
100x−x=−18.18‾−(−0.18‾)100x - x = -18.\overline{18} - (-0.\overline{18})

STEP 5

implify the left side of the equation.
99x=−18.18‾+0.18‾99x = -18.\overline{18} +0.\overline{18}

STEP 6

implify the right side of the equation.
99x=−1899x = -18

STEP 7

Finally, solve for xx by dividing both sides of the equation by99.
x=−1899x = \frac{-18}{99}

STEP 8

We can simplify this fraction by dividing both the numerator and denominator by their greatest common divisor, which is.
x=−18/99/=−211x = \frac{-18/}{99/} = \frac{-2}{11}So, −0.18‾-0.\overline{18} in simplest form is −211-\frac{2}{11}.

STEP 9

Now, let's move on to the second number, 4.45‾4.\overline{45}. We will denote this number as yy.
y=4.45‾y =4.\overline{45}

STEP 10

To get rid of the decimal, we multiply yy by100, because there are two digits in the repeating part.
100y=445.45‾100y =445.\overline{45}

STEP 11

Now, we subtract the original equation from this new equation to get rid of the repeating decimal.
100y−y=445.45‾−4.45‾100y - y =445.\overline{45} -4.\overline{45}

STEP 12

implify the left side of the equation.
99y=445.45‾−4.45‾99y =445.\overline{45} -4.\overline{45}

STEP 13

implify the right side of the equation.
99y=44199y =441

STEP 14

Finally, solve for yy by dividing both sides of the equation by99.
y=44199y = \frac{441}{99}

STEP 15

We can simplify this fraction by dividing both the numerator and denominator by their greatest common divisor, which is9.
y=441/999/9=4911y = \frac{441/9}{99/9} = \frac{49}{11}So, 4.45‾4.\overline{45} in simplest form is 4911\frac{49}{11}.
The solutions are x=−211x = -\frac{2}{11} and y=4911y = \frac{49}{11}.

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